# Swarm Intelligence Algorithms for Weapon-Target Assignment in a Multilayer Defense Scenario: A Comparative Study

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{3}) problem. The proposed schemes were implemented in MATLAB, and PSO was shown to converge rapidly and result in saving more assets with a faster convergence for learning. A novel GA was proposed to solve the sensor-weapon-target assignment problem [30]. Some up-to-date swarm intelligence algorithms have also been developed to solve complex problems, including SCA and the modified backtracking search algorithm (MBSA) [31,32]. Moreover, some swarm intelligence algorithms have been applied in the financial market and image segmentation [33,34].

## 2. Multilayer Defense WTA Model and Benchmark Problem

#### 2.1. WTA Model under Multilayer Defense

_{dsa}: the number of weapons of type d deployed to intercept the offensive weapons of type a to protect asset s (i.e., the defense plan).

_{dsa}is equal to D × S × A. Table 1 lists the other notations used in the model.

#### 2.2. Benchmark Problem and Optimal Solution

_{1}= 100 and B

_{2}= 50.

_{1}= 400, ν

_{2}= 300 and ν

_{3}= 200.

_{11}= 5, n

_{12}= 9, n

_{21}= 25, n

_{22}= 7, n

_{31}= 20 and n

_{32}= 13.

_{1}= 2250, G

_{2}= 1500 and G

_{3}= 1950.

_{1}= 34 and t

_{1}= 51.

_{1}= 20 and C

_{1}= 30, while the total budget C

_{max}= 3800.

_{max1}= 350 and M

_{max2}= 320, respectively, while the manpower required for operating each defending weapon of type 1 and 2 is M

_{1}= 5 and M

_{2}= 4, respectively.

## 3. Solving Methods

- (1)
- Generating the initial condition of each particle, including the initial position in solution space and the initial velocity;
- (2)
- Evaluating the searching point for each particle and updating the global best-so-far position gbest and the local best-so-far position pbest;
- (3)
- Modifying the position and velocity of each particle to obtain a new searching point.

#### 3.1. ACO Algorithm

#### 3.1.1. Coding Scheme

_{dsa}, which is expressed by a row vector:

#### 3.1.2. Definition of Search Space

_{1,1,1}, stage 2 is associated with x

_{1,1,2}, and the last stage L is associated with x

_{D}

_{,S,A}. The ith stage has a total of U

_{i}+1 nodes which are numbered from 0 to U

_{i.}The number on a node represents the specific value being taken by the variable. U

_{i}denotes the maximum possible of the variable at stage i. For the sake of simplicity, we set U

_{i}to B

_{d}, i.e., the number of available defensive weapons of type d. Alternatively, to narrow the search space of a large-scale WTA, while accounting for constraints (2)~(5), U

_{i}is identified by:

#### 3.1.3. Transit Probability

_{ij}(t−1) denotes node j’s pheromone trail at iteration t − 1, $allo{w}_{i}^{t}\left(t\right)$ is an allowable set of nodes for ant k, comprising those nodes of stage i that do not produce any constraint conflict if the decision variable of stage i takes the values on those nodes. Of course, $allo{w}_{i}^{t}\left(t\right)$ will depend on the particular values of the decision variables linked to the previous i − 1 stages. We define $allo{w}_{i}^{t}\left(t\right)$ through a feasibility check, which is conducted in the following way: Given the values for the variables at the previous i − 1 stages, the value of all variables after stage i is set to 0; only the variable x

_{d,s,a}of stage i can change. Then, x

_{d,s,a}is set to 0, 1, 2, …, in turn. For each value taken by x

_{d,s,a}, check whether any constraint is violated. Once a value, say l, leads to some constraint unmet, then the feasibility check is terminated and set $allo{w}_{i}^{t}\left(t\right)=\left\{0,1,2,\dots ,l-1\right\}$. Obviously, the introduction of $allo{w}_{i}^{t}\left(t\right)$ makes the solutions constructed by ants always feasible. In this way, we have managed to manipulate the constraints and drastically reduce the search space.

#### 3.1.4. Pheromone Trail Updating

_{ij}(t).

_{0}denotes the initial value of pheromone trail, f

_{best_so_far}and f

_{average}(t) represents the current maximum objective function and average objective function, respectively. To prevent prematurity, referring to the practice in MMAS, the pheromone trail is limited within a predefined interval [τ

_{min}, τ

_{max}].

#### 3.2. BPSO Algorithm

#### 3.2.1. Coding Scheme

_{dsa}are also arranged in a row vector X that is expressed by (6). Then, each component of X is decoded into a binary substring. In this way, one solution to the problem, which can be thought of as one particle, is represented by one binary string, which consists of D × S × A binary substrings. The length of the total binary string is jointly determined by the length of the integer interval associated with each decision variable and the number of decision variables.

_{dsa}is U

_{d,s,a}. U

_{d,s,a}can be determined by an approach similar to that used in ACO described by Equation (7). Then, the integer interval within which x

_{dsa}can take the value of [0, U

_{d,s,a}]. The encoding length of x

_{dsa}is set to l

_{d,s,a}= [log

_{2}U

_{d,s,a}] + 1 (the expression [log

_{2}U

_{d,s,a}] means the maximum integer that does not exceed log

_{2}U

_{d,s,a}). For example, if U

_{d,s,a}= 50, then l

_{d,s,a}= 6. Therefore, the length (or the number of bits) of a binary string representing one solution (particle) is $\sum}_{d}{\displaystyle \sum}_{s}{\displaystyle \sum}_{a}{l}_{d,s,a$.

#### 3.2.2. Initialization of Particle Population

_{dsa}in a binary system is l

_{d.s.a}. Let each bit of x

_{dsa}’s binary string take 1 with a probability of p

_{d}rather than the usual 0.5 and take 0 with a probability of (1 − p

_{d}). It is not difficult to deduce that the expected value of x

_{dsa}is ${p}_{d}=\left({2}^{{l}_{d,s,a}}-1\right)$. For a given weapon type d, the weapon availability constraint is $\sum}_{s}{\displaystyle \sum}_{a}{x}_{dsa}\le {B}_{d$. Replacing x

_{dsa}in this constraint by its expected value, we have $\sum}_{s}{\displaystyle \sum}_{a}{p}_{d}\left({2}^{{l}_{d,s,a}}-1\right)\le {B}_{d$. This means ${p}_{d}\le \frac{{B}_{d}}{{\displaystyle \sum _{s}{\displaystyle \sum _{a}({2}^{{l}_{d,s,a}}-1)}}}$.

#### 3.2.3. Fitness Measure

#### 3.2.4. Particle Velocity and Position Updating

_{ij}denotes the jth component of the best position that particle i has achieved so far; gbest

_{j}denotes the j-th component of the global best position that the population has achieved so far; r

_{1}and r

_{2}are random numbers between 0 and 1 with uniform distribution; c

_{1}and c

_{2}are accelerating coefficients; and w is an inertial coefficient.

_{ij}is a random number between [0, 1] with a uniform distribution, and $sig\left({V}_{ij}^{k}\right)=1/(1+\mathrm{exp}\left(-{V}_{ij}^{k}\right))$ is the sigmoid function. To avoid $sig\left({V}_{ij}^{k}\right)$ approaching too closely to 0 or 1, we set ${V}_{ij}^{k}$ within the range of [−6, +6].

_{0}(0 < c

_{0}< 1). For each particle, when updating its velocity and position, a re-initialization check is conducted. To do so, first, a random number r between [0, 1] with uniform distribution is generated. Then, r and c

_{0}are compared. If r > c

_{0}, then formula Equations (12) and (13) are used to update the particle’s velocity and position; otherwise, its position is reinitialized using the method described above, and the velocity is reset to 0.

#### 3.3. IPSO Algorithm

#### 3.3.1. Coding Scheme

_{dsa}takes its actual integer value within the interval [0,U

_{d,s,a}]. One vector is a particle representing one solution to the problem. Obviously, the encoding length of one complete solution is the same as the number of components in X.

#### 3.3.2. Initialization of Particle Population

_{d,s,a}] with uniform distribution for component x

_{dsa}, a large number of infeasible solutions will be produced. Hence, it is necessary to exert additional control on the particle population initialization. Most likely, the simplest control method is to compress the initial position vector. Note that the expected value of x

_{dsa}randomly generated within the interval [0,U

_{d,s,a}] is U

_{d,s,a}/2. For a given weapon type d, assume the compression factor for component x

_{dsa}is η

_{d}. Once a specific value for x

_{dsa}is produced, then compress it to η

_{d}x

_{dsa}. The expected value for this compressed variable is reduced to η

_{d}U

_{d,s,a}/2 with the same proportion. Next, replace x

_{dsa}in the weapon availability constraint (Equation (2)) by this compressed expected value, and we have $\sum}_{s}{\displaystyle \sum}_{a}{\eta}_{d}{U}_{d,s,a}/2\le {B}_{d$. Hence, ${\eta}_{d}\le \frac{2{B}_{d}}{{{\displaystyle \sum}}_{s}{{\displaystyle \sum}}_{a}{U}_{d,s,a}}$. For the sake of simplifying the calculations, we set:

_{dsa}, we multiply x

_{dsa}by the associated compression factor η

_{d}. Then, the product η

_{d}X

_{dsa}is rounded off and is used as a substitute for x

_{dsa}. Nearly half of the population will meet the weapon availability constraint. In this way, a balance can be obtained between the diversification of the initial population and a better iteration starting point.

#### 3.3.3. Fitness Measure

#### 3.3.4. Particle Velocity and Position Updating

_{ij}denotes the j-th component of the best position that particle i has achieved so far; gbest

_{j}denotes the j-th component of the global best position that the population has achieved so far; r

_{1}and r

_{2}are random numbers between 0 and 1 with uniform distribution; c

_{1}and c

_{2}are accelerating coefficients; and $w$ is an inertial coefficient. The symbol (roundoff) denotes taking an integer number that is the closest to a given real number.

- (a)
- If ${X}_{ij}^{k}<0$, then set ${X}_{ij}^{k}=0$ and ${V}_{ij}^{k}=0$;
- (b)
- If ${X}_{ij}^{k}>{U}_{j}$, then set ${X}_{ij}^{k}={U}_{j}$ and ${V}_{ij}^{k}=0$.

_{j}represents the maximum possible for the decision variable associated with the j-th component of the position vector.

_{0}(0 < c

_{0}< 1) is set. When updating a particle, a random number r between [0, 1] with uniform distribution is generated. If r ≤ c

_{0}, then the particle position to be updated is reinitialized, and the velocity is reset to 0; otherwise, formula Equations (15) and (16) are used to update the velocity and position.

#### 3.4. SCA Algorithm

_{dsa}in SCA does not address velocity updating. The updating rule in SCA is:

_{1}, r

_{2}, r

_{3}and r

_{4}are random numbers, and ${P}_{i}^{t}$ is the position of the destination point in the i-th dimension at iteration t. ${X}_{i}^{t+1}$ is updated by Equation (17). r

_{1}is calculated by:

_{3}from a random value to a fixed value of 0.9. Second, the original SCA only exploits the current optimal solution, while we change ${X}_{i}^{t}$ in the second formula of Equation (17) to the global optimal position of particles in the i-th dimension.

## 4. Performance Evaluation Based on a Benchmark Problem

#### 4.1. Solving Results

_{0}= 1; evaporation coefficient of pheromone trail ρ = 0.01; maximum pheromone trail τ

_{max}= 10τ

_{0}= 1; minimum pheromone trail τ

_{min}= 0.1τ

_{0}; maximum iteration number iter

_{max}= 1000 (stopping criterion).

_{1}= c

_{2}= 2; re-initialization probability c

_{0}= 0.1; penalty factor α = 100; maximum iteration number iter

_{max}= 1000 (stopping criterion).

#### 4.2. Discussion

#### 4.2.1. Algorithm Effectiveness

#### 4.2.2. Comparison of Algorithms

## 5. Solving a Large-Scale WTA Problem Using IPSO

_{1}= 560, B

_{2}= 300 and B

_{3}= 140. The manpower needed per defensive weapon of types 1, 2 and 3 are m

_{1}= 6, m

_{2}= 5 and m

_{3}= 4. The total available manpower for the three types of weapons is M

_{1}= 3500, M

_{2}= 1600 and M

_{3}= 500. The costs of operation and maintenance are c

_{1}= 20, c

_{2}= 30 and c

_{3}= 40. The available budget is C

_{max}= 25,000. The ground area occupied by individual defensive weapons is t

_{1}= 32, t

_{2}= 48 and t

_{3}= 72. Other data, including the asset values, space available and intercept probabilities can be found in [35].

_{1}= 275, R

_{2}= 170 and R

_{3}= 95. The attacking plan and destruction probabilities of various offensive weapons can be found in [35]. Assume that the information about enemy’s attacking plan and destruction probabilities is symmetric to the defender.

#### 5.1. Solving Results

#### 5.2. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**Result summary of the large-scale WTA. Note: avg = average objective function; cv = coefficient of variation; sd = standard deviation of objective function; CPU = average CPU time.

ν_{s}: Value of asset s | G_{s}: Ground area available at asset s |

k_{dsa}: Probability of successful interception by one defending weapon of type d deployed to defend an asset s against an attacking weapon of type a (effectiveness) | c_{d}: Cost of operating one defending weapon of type d |

n_{sa}: Number of attacking weapons of type a aimed at asset s (attack plan) | C_{max}: Maximum operating cost of weapons deployed |

g_{sa}: Probability that a single offensive weapon of type a can penetrate the defense system and destroy asset s (probability of damage) | m_{d}: Manpower required for operating each defensive weapon of type d |

B_{d}: Number of defending weapons of type d | M_{d}: Maximum available manpower to operate defending weapons of type d |

t_{d}: Ground area occupied by deploying each defensive weapon of type d |

ACO | BPSO | IPSO | SCA | |||||
---|---|---|---|---|---|---|---|---|

Objective Function | Relative Deviation | Objective Function | Relative Deviation | Objective Function | Relative Deviation | Objective Function | Relative Deviation | |

1 | 0.59392 | −0.556% | 0.58407 | −2.205% | 0.59713 | −0.018% | 0.57176 | −4.27% |

2 | 0.59159 | −0.946% | 0.57480 | −3.757% | 0.58147 | −2.640% | 0.59194 | −0.89% |

3 | 0.59617 | −0.179% | 0.58557 | −1.954% | 0.59651 | −0.122% | 0.56355 | −5.64% |

4 | 0.59248 | −0.797% | 0.58064 | −2.779% | 0.59722 | −0.003% | 0.55336 | −7.35% |

5 | 0.59009 | −1.197% | 0.58761 | −1.612% | 0.58891 | −1.395% | 0.55544 | −7.00% |

6 | 0.59325 | −0.668% | 0.58910 | −1.363% | 0.57674 | −3.432% | 0.56413 | −5.54% |

7 | 0.59135 | −0.986% | 0.58917 | −1.351% | 0.59724 | 0 | 0.55186 | −7.60% |

8 | 0.59561 | −0.273% | 0.56630 | −5.180% | 0.59724 | 0 | 0.57834 | −3.16% |

9 | 0.59497 | −0.380% | 0.57925 | −3.012% | 0.59688 | −0.060% | 0.56639 | −5.17% |

10 | 0.59547 | −0.296% | 0.57915 | −3.029% | 0.57189 | −4.245% | 0.56320 | −5.70% |

Best | 0.59617 | −0.179% | 0.58917 | −1.351% | 0.59724 | 0 | 0.59194 | −0.89% |

Worst | 0.59009 | −1.197% | 0.56630 | −5.180% | 0.57189 | −4.245% | 0.55186 | −7.60% |

Average | 0.59349 | −0.628% | 0.58157 | −2.624% | 0.59012 | −1.192% | 0.56600 | −5.23% |

standard deviation | 0.00208 | 0.00718 | 0.00986 | 0.01223 | ||||

coefficient of variation | 0.350% | 1.235% | 1.671% | 2.16% | ||||

average CPU time | 1362 ms | 456 ms | 95 ms | 120 ms |

Solution | |
---|---|

ACO (0.59617) | ${x}_{111}$ = 0, ${x}_{112}$ = 43, ${x}_{121}$ = 2, ${x}_{122}$ = 0, ${x}_{131}$ = 25, ${x}_{132}$ = 0 ${x}_{211}$ = 6, ${x}_{212}$ = 0, ${x}_{221}$ = 28, ${x}_{222}$ = 0, ${x}_{231}$ = 0, ${x}_{232}$ = 16 |

BPSO (0.58917) | ${x}_{111}$ = 0, ${x}_{112}$ = 53, ${x}_{121}$ = 2, ${x}_{122}$ = 0, ${x}_{131}$ = 15, ${x}_{132}$ = 0 ${x}_{211}$ = 4, ${x}_{212}$ = 0, ${x}_{221}$ = 26, ${x}_{222}$ = 2, ${x}_{231}$ = 8, ${x}_{232}$ = 10 |

IPSO (0.59724) | identical to the theoretical optimal solution |

SCA (0.59194) | ${x}_{111}$ = 0, ${x}_{112}$ = 44, ${x}_{121}$ = 0, ${x}_{122}$ = 0, ${x}_{131}$ = 26, ${x}_{132}$ = 0 ${x}_{211}$ = 11, ${x}_{212}$ = 0, ${x}_{221}$ = 28, ${x}_{222}$ = 0, ${x}_{231}$ = 0, ${x}_{232}$ = 11 |

Defending Weapon d | Asset s | $\mathbf{Variable}\text{}{\mathit{x}}_{\mathit{d}\mathit{s}\mathit{a}}$ | Defending Weapon d | Asset s | $\mathbf{Variable}\text{}{\mathit{x}}_{\mathit{d}\mathit{s}\mathit{a}}$ | Defending Weapon d | Asset s | $\mathbf{Variable}\text{}{\mathit{x}}_{\mathit{d}\mathit{s}\mathit{a}}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Attacking Weapon a | Attacking Weapon a | Attacking Weapon a | ||||||||||||

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | ||||||

1 | 1 | 2 | 16 | 6 | 2 | 1 | 13 | 8 | 13 | 3 | 1 | 5 | 3 | 3 |

1 | 2 | 21 | 17 | 11 | 2 | 2 | 13 | 1 | 4 | 3 | 2 | 1 | 0 | 3 |

1 | 3 | 10 | 2 | 8 | 2 | 3 | 6 | 5 | 7 | 3 | 3 | 4 | 2 | 5 |

1 | 4 | 6 | 17 | 19 | 2 | 4 | 0 | 16 | 9 | 3 | 4 | 1 | 0 | 1 |

1 | 5 | 2 | 5 | 8 | 2 | 5 | 3 | 11 | 3 | 3 | 5 | 2 | 3 | 4 |

1 | 6 | 20 | 6 | 17 | 2 | 6 | 0 | 9 | 0 | 3 | 6 | 0 | 4 | 2 |

1 | 7 | 12 | 8 | 1 | 2 | 7 | 7 | 6 | 5 | 3 | 7 | 2 | 6 | 0 |

1 | 8 | 12 | 4 | 5 | 2 | 8 | 9 | 7 | 0 | 3 | 8 | 2 | 2 | 2 |

1 | 9 | 12 | 7 | 10 | 2 | 9 | 7 | 5 | 1 | 3 | 9 | 0 | 6 | 2 |

1 | 10 | 5 | 5 | 12 | 2 | 10 | 9 | 6 | 15 | 3 | 10 | 3 | 3 | 1 |

1 | 11 | 9 | 9 | 9 | 2 | 11 | 7 | 3 | 6 | 3 | 11 | 2 | 0 | 2 |

1 | 12 | 11 | 4 | 11 | 2 | 12 | 0 | 1 | 3 | 3 | 12 | 5 | 0 | 1 |

1 | 13 | 9 | 18 | 18 | 2 | 13 | 10 | 15 | 0 | 3 | 13 | 0 | 1 | 1 |

1 | 14 | 17 | 0 | 6 | 2 | 14 | 8 | 0 | 7 | 3 | 14 | 6 | 3 | 0 |

1 | 15 | 11 | 12 | 6 | 2 | 15 | 0 | 0 | 4 | 3 | 15 | 2 | 4 | 2 |

1 | 16 | 0 | 9 | 17 | 2 | 16 | 9 | 4 | 0 | 3 | 16 | 0 | 3 | 0 |

1 | 17 | 17 | 21 | 3 | 2 | 17 | 1 | 1 | 0 | 3 | 17 | 0 | 1 | 5 |

1 | 18 | 0 | 0 | 4 | 2 | 18 | 0 | 6 | 0 | 3 | 18 | 1 | 0 | 1 |

1 | 19 | 7 | 10 | 8 | 2 | 19 | 7 | 1 | 4 | 3 | 19 | 3 | 3 | 1 |

1 | 20 | 0 | 12 | 5 | 2 | 20 | 1 | 4 | 0 | 3 | 20 | 1 | 0 | 5 |

$\mathbf{Defensive}\text{}\mathbf{Weapon}\text{}\mathit{d}$ | Number Deployed | Manpower Required |
---|---|---|

1 | 549 | 3294 |

2 | 300 | 1500 |

3 | 125 | 500 |

Asset s | Space Occupied | Asset s | Space Occupied |
---|---|---|---|

1 | 3192 | 11 | 1920 |

2 | 2720 | 12 | 1456 |

3 | 2296 | 13 | 2784 |

4 | 2688 | 14 | 2104 |

5 | 1944 | 15 | 1696 |

6 | 2240 | 16 | 1672 |

7 | 2112 | 17 | 1840 |

8 | 1872 | 18 | 560 |

9 | 2128 | 19 | 1880 |

10 | 2648 | 20 | 1216 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cao, M.; Fang, W.
Swarm Intelligence Algorithms for Weapon-Target Assignment in a Multilayer Defense Scenario: A Comparative Study. *Symmetry* **2020**, *12*, 824.
https://doi.org/10.3390/sym12050824

**AMA Style**

Cao M, Fang W.
Swarm Intelligence Algorithms for Weapon-Target Assignment in a Multilayer Defense Scenario: A Comparative Study. *Symmetry*. 2020; 12(5):824.
https://doi.org/10.3390/sym12050824

**Chicago/Turabian Style**

Cao, Ming, and Weiguo Fang.
2020. "Swarm Intelligence Algorithms for Weapon-Target Assignment in a Multilayer Defense Scenario: A Comparative Study" *Symmetry* 12, no. 5: 824.
https://doi.org/10.3390/sym12050824